As they fly past, the Earth tries to slow them down, but Earth and this object are moving in the same direction, so the amount of push we can give an object heading out toward the outer solar system, that's going in the same direction of orbit we're going in, is a lot more than the pull we give it as it goes past us.

I'm pretty sure that the direction they're talking about is the planet's and spacecraft's motion around the sun, not the planet's rotation around its axis.